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diff --git a/projects/projectile/index.html b/projects/projectile/index.html index 63d3938..a27b692 100644 --- a/projects/projectile/index.html +++ b/projects/projectile/index.html @@ -34,17 +34,19 @@ <br /><br /><br /> <h3>Projectile Motion</h3> -<p>To determine to position of the projectile we should compute the position vector \(\vec{r}(t)=x(t)\vec{i}+y(t)\vec{i}\).</p> +<p>To determine to position of the projectile we should compute the position vector \(\vec{r}(t)=x(t)\vec{i}+y(t)\vec{j}\).</p> <h5>\(x(t)\):</h5> <p>We know from Newton second law that \(\sum \vec{F} = m\times \vec{a}_x = m\times a_x(t)\vec{i}\)</p> <p>However, the projectile as a constant speed along \(\vec{i}\). Hence, \(a_x(t) = 0 \).</p> <p>Thus:</p> +\[ v_x(t) = v_{x,0} \] \[ x(t) = \int_{t_0}^t v_{0,x}dt = v_{0,x}t + C = v_{0,x}t + x_0\] <h5>\(y(t)\):</h5> -<p>We know from Newton second law that \(\sum \vec{F} = m\times \vec{a}_y = m\times a_y(t)\vec{i}\)</p> +<p>We know from Newton second law that \(\sum \vec{F} = m\times \vec{a}_y = m\times a_y(t)\vec{j}\)</p> <p>The projectile is under the influence of the gravity that is oriented <em>downward</em>. Hence, \(a_y(t) = -g \).</p> <p>Thus:</p> \[ v_y(t) = \int_{t_0}^t a_{y}(t)dt = -gt+C = -gt + v_{0,y}\] \[ y(t) = \int_{t_0}^t v_y(t)dt = -\frac{1}{2}gt^2 + v_{0,y}t+C=-\frac{1}{2}gt^2 + v_{0,y}t+y_0\] <h5>\(\vec{r}(t)\):</h5> -Finally knowing \(x(t)\) and \(y(t)\) we have \( \vec{r}(t) = \left(\begin{smallmatrix}x(t)\\y(t)\end{smallmatrix}\right) = \left(\begin{smallmatrix}v_{0,x}t + x_0\\-\frac{1}{2}gt^2 + v_{0,y}t+y_0\end{smallmatrix}\right)\) +<p>Finally knowing \(x(t)\) and \(y(t)\) we have \( \vec{r}(t) = \left(\begin{smallmatrix}x(t)\\y(t)\end{smallmatrix}\right) = \left(\begin{smallmatrix}v_{0,x}t + x_0\\-\frac{1}{2}gt^2 + v_{0,y}t+y_0\end{smallmatrix}\right)\)</p> +<p>We can deduce also that \( \vec{v}(t) = \left(\begin{smallmatrix}v_x(t)\\v_y(t)\end{smallmatrix}\right) = \left(\begin{smallmatrix}v_{0,x}\\-gt+v_{0,y}\end{smallmatrix}\right)\)</p> |